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Graph signal processing and wavelet packets
No full textNowadays graphs became of significant importance given their use to describe complex system dynamics, with important applications to real world problems, e.g. graph representation of the brain, social networks, biological networks, spreading of a disease, etc..
What is missing in graph signal processing is a general definition of Spectral Graph Wavelet Packets Transform in the same fashion as for the classical framework where the equivalent of frequency is represented by the eigenvalues of the Laplacian matrix. Bremer and coauthors introduced diffusion wavelet packets transforms starting from diffusion wavelet definition, based on a diffusion operator
T on a manifold or a graph. Cloninger et al., defined the natural graph wavelet packet dictionaries by introducing a set of novel multiscale basis transforms by considering the distance between graph Laplacian eigenvectors.
In this paper we introduce a novel graph wavelet packets construction, to our knowledge different from the ones known in literature. Our work is inspired by the Spectral Graph Wavelet Transform (SGWT) defined by Hammond et al., and can be
viewed as a generalization of their work. The result is a dictionary of frames particularly suitable for analyzing signals defined on graphs with a large number of nodes.
We will give some concrete examples on how the wavelet packets can be used for compressing, denoising and reconstruction by considering a signal, given by the fRMI (functional magnetic resonance imaging) data, on the nodes of voxel-wise brain graph
G with 900.760 nodes (representing the brain voxels)
Exploiting Ultrasensitivity for Biomolecular Implementation of a Control System without Error Detection
No full textGraph signal processing and wavelet packets .
No full textNowadays graphs became of significant importance given their use to describe complex system dynamics, with important applications to real world problems, e.g. graph representation of the brain, social networks, biological networks, spreading of a disease, etc..
What is missing in graph signal processing is a general definition of Spectral Graph Wavelet Packets Transform in the same fashion as for the classical framework where the equivalent of frequency is represented by the eigenvalues of the Laplacian matrix. Bremer and coauthors introduced diffusion wavelet packets transforms starting from diffusion wavelet definition, based on a diffusion operator
T on a manifold or a graph. Cloninger et al., defined the natural graph wavelet packet dictionaries by introducing a set of novel multiscale basis transforms by considering the distance between graph Laplacian eigenvectors.
In this paper we introduce a novel graph wavelet packets construction, to our knowledge different from the ones known in literature. Our work is inspired by the Spectral Graph Wavelet Transform (SGWT) defined by Hammond et al., and can be
viewed as a generalization of their work. The result is a dictionary of frames particularly suitable for analyzing signals defined on graphs with a large number of nodes.
We will give some concrete examples on how the wavelet packets can be used for compressing, denoising and reconstruction by considering a signal, given by the fRMI (functional magnetic resonance imaging) data, on the nodes of voxel-wise brain graph
G with 900.760 nodes (representing the brain voxels)
Community structure of the Italian Covidome.
No full textA. The Covidome partition, after consensus clustering, for the hospitalized with symptoms time series, represented on the map (left panel) and on the Covidome graph (right panel). B. The Covidome partition, after consensus clustering, for the new positives time series, on the map (left panel) and on the Covidome graph (right panel). C. The Covidome allegiance matrix (left panel) across the six different Covid indicators (i.e., number of hospitalized individuals in ICU, number of hospitalized individuals with symptoms, number of individuals in home isolation, new positives, discharged healed and deceased individuals, respectively). The representation of the Northern (blue) and Southern (red) modules from the allegiance matrix and of the swing regions (green), respectively, on the Italian map (central panel) and on the graph (right panel). Notice that we have chosen the interval [0, 1] for the Covidome because all the values are positive.</p
Maximum and minimum values for the time windows <i>W</i><sub>1</sub> = 1th March-9th April and <i>W</i><sub>2</sub> = 26th October-4th December corresponding to a range of 10 days before the first and second lockdowns and 30 days after, respectively, for mean dynamic Covidome time series (Fig 3).
No full textHS–hospitalized with symptoms, NP–new positives.</p
Workflow of the Covid connectivity analysis for hospitalized individuals with symptoms in Italy, during 2020.
No full textA. The time series of hospitalized individuals with symptoms for all the 20 Italian regions. B. The Covidome (the adjacency matrix of the network) obtained by computing the Pearson’s correlation coefficients associated to data reported in panel A. C. Communities of the Covidome for the considered time series, represented both on the Italian map (left panel) and on the graph (right panel), respectively. D. Average Covid connectivity obtained using sliding window correlation. The three different curves represent three different areas corresponding to Northern, Central and Southern Italy.</p
New positives dynamic Covidome.
No full textOn the top row we have represented the dynamic Covidome (left panel) and the Italian regions map (right panel) containing the normalized (to [0, 1] interval) regional average connectivity of the dynamic Covidome in time. On the bottom row we have plotted the three time series corresponding to the mean dynamic Covidomes for the aforementioned Italian areas. The green sliding window depicts the 21 days time window. (AVI)</p
Optimal control of monomers and oligomers degradation in an Alzheimer’s disease model
No full textThe aggregation and accumulation of oligomers of misfolded A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}-amyloids in the human brain is one of the possible causes for the onset of the Alzheimer's disease in the early stage. We introduce and study a new ODE model for the evolution of Alzheimer's disease based on the interaction between monomers, proto-oligomers, and oligomers of A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} amyloid protein in a small portion of the human brain, based upon biochemical processes such as polymerization, depolymerization, fragmentation and concatenation. We further introduce the possibility of controlling the evolution of the system via a treatment that targets the monomers and/or the oligomers. We observe that a combined optimal treatment on both monomers and oligomers induces a substantial decrease of the oligomer concentration at the final stage. A single treatment on oligomers performs better than a single treatment on monomers. These results shed a light on the effectiveness of immunotherapy using anti-A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} antibodies, targeting monomers or oligomers. Several numerical simulations show how the oligomer concentration evolves without treatment, with single monomer/oligomer treatment, or with a combined treatment
Covidome and structural connectome.
No full textA. Dynamic Covidomes (top row) and structural connectome for the geographical distance between Italian regions for hospitalized individual with symptoms. B. Time series correlation between three different sub-matrices of dynamic Covidome and structural connectome, respectively, corresponding to Northern, Central and Southern Italy for the Covid indicator introduced in A. C. Dynamic Covidomes (top row) and structural connectome for the geographical distance between Italian regions for new positives. D. Time series correlation between three different sub-matrices of dynamic Covidome and structural connectome, respectively, corresponding to Northern, Central and Southern Italy for the Covid indicator introduced in C. Notice the different range in the Covid connectivity between the tow indicators.</p
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